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<HTML>
<HEAD>
<TITLE>Michel Lavrauw's Home Page</TITLE>
</HEAD>
<BODY>
<H1><center>Michel Lavrauw</center></H1>
<hr />
<center>
<a href="index.html">Home</a> |
<a href="research.html">Research</a> |
<!--<a href="events.html">Events</a> |-->
<a href="teaching.html">Teaching</a> |
<a href="fining.html">FinInG</a> |
<!--<a href="bio.html">BIO</a> |-->
<!--<a href="cv.html">CV</a> |-->
<a href="picture.html">Picture</a>
</center>
<hr />
<h3>TEACHING</h3>
<ul>
<li><a href="#AG_Selected_Topics_Algebra"><i>Algebraic Geometry - Selected Topics in Algebra (1)</i></a> (University of Primorska) (2025)</li>
<li><a href="https://mlavrauw.github.io/GapDaysSpring2025/gapdays_spring2025.html"><i>Projective Geometry, Coding Theory, Incidence Geometry in GAP</i></a>, <i>(Vrije Universiteit Brussel) (2025)</i>
<li><a href="#Galois_Theory"><i>Galois Theory</i></a> (University of Primorska) (2023, 2024, 2025)</li>
<li><a href="#AC_Selected_Topics_Algebra"><i>Algebraic Curves - Selected Topics in Algebra (1)</i></a> (University of Primorska) (2023, 2024)</li>
<li><a href="#Algebra_IV"><i>Algebra IV - Rings and Fields</i></a> (University of Primorska) (2023, 2024, 2025, 2026)</li>
<li><a href="#Coding_Theory"><i>Coding Theory</i></a> (University of Primorska) (2023, 2024, 2025, 2026)</li>
<li> <a href="#Algebra_I"><i>Algebra I - Groups and Rings</i></a> (Sabancı University) (Fall 2015, Fall 2018, Fall 2022)</li>
<li> <a href="#Algebra_II"><i>Algebra II - Modules, Fields, and Galois Theory</i></a> (Sabancı University) (Spring 2019, Spring 2023)</li>
<li> <a href="#DM"><i>Discrete Mathematics</i></a> (Sabancı
University) (Spring 2017, Spring 2018, Fall 2021, Fall 2022)</li>
<li> <a href="#CT"><i>Introduction to Coding Theory</i></a> (Sabancı University) (Spring 2021)</li>
<li> <a href="#AG"><i>Introduction to Algebraic Geometry</i></a> (Sabancı University) (Fall 2018)</li>
<li> <a href="#PG"><i>Projective Geometry</i></a> (Sabancı
University) (Fall 2016, Fall 2017, Fall 2019, Fall 2020, Spring 2022)</li>
<li> <a href="#CC"><i>Coding Theory and Cryptography</i></a>
(University of Canterbury, Christchurch, New Zealand) (September 2019)</li>
<li> <a href="#Intro_Algebra"><i>Introduction to Algebra</i></a>
(Sabancı University) (Spring 2019, Spring 2020, Spring 2021, Spring 2022)</li>
<li> <a href="#FG"><i>Finite Geometry</i></a> (Sabancı University) (Fall 2015, Spring 2020)</li>
<li> <a href="#LA"><i>Linear Algebra</i></a> (Sabancı University) (Fall 2016, Fall 2017)</li>
<li> <a href="#AC_village"><i>Algebraic Curves</i></a>
(<a href="https://en.wikipedia.org/wiki/Nesin_Mathematics_Village">Mahematics Village</a>) (2019)</li>
<li> <a href="#IGB"><i>Incidence Geometry and Buildings</i></a>
(<a href="https://en.wikipedia.org/wiki/Nesin_Mathematics_Village">Mahematics
Village</a>) (2019) (<a href="inc_geom_buildings_notes.pdf">notes</a>)</li>
<li> <a href="#PG_village"><i>Projective Geometry</i></a>
(<a href="https://en.wikipedia.org/wiki/Nesin_Mathematics_Village">Mahematics
Village</a>) (2018, 2019) (<a href="pg_notes_village.pdf">notes</a>)</li>
<li> <a href="#FALG"><i>Fondamenti di Algebra Lineare e Geometria</i></a> (University of Padua) (2011, 2012, 2013, 2014, 2015, 2016) (in Italian)</li>
<li> <a href="#LAFF"><i>Linear Algebra and Finite Fields</i></a> (Ghent University) (2007, 2008, 2009, 2010, 2011)<br/></li>
<li> <a href="#PG_Ghent"><i>Projectieve Meetkunde</i></a> (Ghent University) (2007, 2008, 2009, 2010) (in Dutch)<br/></li>
</ul>
<p><a href="#teaching_experience">Here</a> is a list of my teaching experience during my PhD at Eindhoven University of Technology.
<p>You can also find some <a href="#gentlessen">notes</a> on <i>Finite semifields and related structures in finite geometry</i>, which I wrote in 2006 for a few lectures I gave as part of the course <i>Finite Geometry</i> at Ghent University.
<hr />
<!--
<center>
<code>
<a href="#LAFF">LAFF</a> |
<a href="#PG">PG</a>
|
<a href="#Ba_Projects">Bachelor Projects</a> |
<a href="#Ma_Projects">Master Projects
</code>
</center>
<hr />
-->
<p><a href="#top"><small>back to top</small></a>
<hr />
<h2><code><a name="AG_Selected_Topics_Algebra">Algebraic Geometry - Selected Topics in Algebra (1) (University of Primorska)</code></h2>
<p>Graduate course</p>
<p>Content: Affine and projective algebraic sets, ideals and coordinate rings, irreducibility and decomposition into irreducible components, Zariski topology, Nullstellensatz (affine and projective), regular and rational functions, regular and rational maps, local rings and maximal ideals, tangent spaces and nonsingular points, local parameters, dimension of varieties, finite maps, normal varieties and normalisation of curves, blowups and resolution of plane curve singularities, examples of projective varieties.
<p> Objectives: The students become acquainted with the foundations of modern algebraic geometry, with an emphasis on the intrinsic, coordinate-free viewpoint and on the relation between geometric properties of varieties and the algebraic properties of their coordinate rings.
<p><a href="#top"><small>back to top</small></a>
<hr />
<h2><code><a name="Galois_Theory">Galois Theory (University of Primorska)</code></h2>
<p>Undergraduate course</p>
<p>Content: - Classical formulas
- Polynomials and field theory
- Fundamental theorem on symmetric polynomials and discriminants
- Roots of unity and cyclotomic polynomials
- Solvability by radicals
- Basic elements of Galois theory. Automorphisms. Galois extensions
- Fundamental theorem of Galois theory
- Classical theorems by Abel, Galois, Gauss, Kronecker, Lagrange, and Ruffini
- Applications and examples
<p>Objectives: The student gets to know the historical context and the modern approach to the problem of solving polynomial equations with radicals.
<p>General competences: The student learns mathematical thinking and recognizes the rigorous mathematical language. In this way he achieves a sovereign and a critical attitude towards different ways of addressing specific mathematical content.
<p>Course-specific competences: The student learns the foundations and modern applications of Galois Theory, widely regarded as one of the most elegant areas of mathematics.
<p> Prerequisites: Analysis I - Foundations of Analysis, Algebra I - Matrix Calculus, Algebra II - Linear Algebra, Algebra III - Abstract Algebra, Discrete Mathematics I - Set Theory.
<p><a href="#top"><small>back to top</small></a>
<hr />
<h2><code><a name="AC_Selected_Topics_Algebra">Algebraic Curves - Selected Topics in Algebra (1) (University of Primorska)</code></h2>
<p>Graduate course</p>
<p>Content: Algebraic sets, ideals, Hilbert basis Theorem, irreducibility, nullstellensatz, coordinate rings, polynomial maps, rational functions, local rings, multiplicity, tangent lines, intersection numbers, affine and projective space, plane curves, Bezout's theorem, varieties, dimension of varieties, resolution of singularities, Riemann-Roch Theorem.
<p> Objectives: the students become acquainted with more advanced concepts of algebra.
<p><a href="#top"><small>back to top</small></a>
<hr />
<h2><code><a name="Algebra_IV">Algebra IV - Rings and Fields (University of Primorska)</code></h2>
<p>Undergraduate course</p>
<p>Content:
- Rings - Ideals - Homomorphism of rings - Factor rings - Integral domain - Euclidean rings -
Principal ideal rings - Gaussian integers - Chinese remainder theorem - Unique factorisation domains - Euclidean domains - Polynomial rings - Irreducibility criteria
- Fields - Subfields - Extensions - Finite extension - Finite fields - The degree theorem - Algebraic extensions - Splitting field - Constructions with ruler and compass - Squaring the circle - Angle trisection - Doubling Cube - Constructions of regular polygons - Primitive element theorem - Basics of Galois theory.
<p><a href="#top"><small>back to top</small></a>
<hr />
<h2><code><a name="Coding_Theory">Coding Theory (University of Primorska)</code></h2>
<p>Undergraduate course</p>
<p>Content:
- mathematical background (groups, rings, ideals, vector spaces, finite fields);
- basic concepts in coding theory;
- algebraic methods for the construction of error correcting codes;
- Hamming codes;
- Linear codes;
- Binary Golay codes;
- Cyclic codes;
- BCH codes;
- Reed-Solomon codes;
- bounds (Hamming, Singleton, Johnson's bound , ...)
<p><a href="#top"><small>back to top</small></a>
<hr />
<h2><code><a name="Algebra_I">Algebra I - Groups and Rings (Sabancı University)</code></h2>
<p>Graduate course for math students</p>
<p>Contents:</p>
<p> Groups: definition and examples, homomorphisms, subgroups, normal subgroups, quotient groups, cyclic groups, symmetric group, group actions, Sylow subgroups, direct sums, free abelian groups, finitely generated abelian groups, free groups, solvable groups, finite simple groups.</p>
<p> Rings: definitions and examples, homomorphisms, ideals, commutative rings, factorization, polynomial rings, formal power series, groups rings, localization, Groebner bases.</p>
<p> All theory will be build from scratch, and a basic (linear) algebra course and some prior experience with abstract mathematics should allow a student to follow the course.
Almost all material which will be treated in the course can be found in any graduate level algebra book, some example of which are: [1] Abstract algebra, by Dummit and Foote; [2] Algebra, by Hungerford; [3] Algebra, by Lang.
<p> This is the first part of the two-semester basic algebra course for graduate students. The aim is to strengthen students' familiarity with basic algebraic structures which are commonly used in all parts of mathematics. These structures include groups, rings, vector spaces, modules and fields.
<p><a href="#top"><small>back to top</small></a>
<hr />
<h2><code><a name="Algebra_II">Algebra II - Modules, Fields, and Galois Theory (Sabancı University)</code></h2>
<p>Graduate course</p>
<p>Contents: modules over principal ideal domains, algebraic field extensions, splitting fields, algebraic closures, separable and inseparable extensions, cyclotomic extensions, automorphisms, Galois theory, finite fields.</p>
<p>This is the second part of the two-semester basic algebra course for beginning graduate students. The aim is to strengthen students' familiarity with basic algebraic structures which are commonly used in all parts of mathematics. These structures are groups, rings, modules and fields.</p>
<p>In Algebra I, the focus lies on groups and rings. Algebra II deals with modules, fields and Galois theory.</p>
<p><a href="#top"><small>back to top</small></a>
<hr />
<h2><code><a name="AG">Introduction to Algebraic Geometry (Sabancı University)</code></h2>
<p>Graduate course</p>
<p>Contents: Projective and affine algebraic varieties, Zariski topology, projective equivalence, normal rational curves, Veronese varieties, Segre varieties, coordinate rings, product varieties, graph of a regular map, cones and projections, classification of quadrics, rational function fields, rational maps, birational equivalence, Noether's normalisation theorem, Hilbert's nullstellensatz, irreducible decomposition, Grassmann varieties, Plücker coordinates, the Klein correspondence, tangent spaces, singular locus of a variety, dimension and degree.</p>
<p>This course aims to prepare the students for more advanced courses on algebraic geometry. The students are introduced to the basic notions of algebraic varieties, and their associated coordinate rings and functions fields. Classical examples of algebraic varieties are studied. The course includes a detailed analysis of the correspondences between algebra and geometry, highlighting the fundamental ideas behind algebraic geometry.</p>
<p>The course is intended to allow the students to develop a good algebraic-geometric intuition before engaging in more abstract algebraic geometry.</p>
<hr />
<h2><code><a name="AG">Introduction to Coding Theory (Sabancı University)</code></h2>
<p>Graduate course</p>
<p>Contents: The course covers basic theory of error-correcting codes.</p>
<p><a href="#top"><small>back to top</small></a>
<hr />
<h2><code><a name="DM">Discrete Mathematics (Sabancı University)</code></h2>
<p>Undergraduate course for engineering and computer science students</p>
<p>Contents: Foundations and basic structures.
Algorithms, growth of functions.
Number theory and cryptography.
Sequences, summation, induction, recursion.
Counting, solving recurrence relations.
Relations, graphs and trees.
Applications.
<p>This course aims to introduce basic ideas of discrete mathematics and formal mathematical reasoning techniques. The course gives students training to develop their mathematical skills, analytical and critical thinking abilities, their ability to apply these capabilities to practical problems, and to communicate their knowledge of these areas. Part of the theory is illustrated using the mathematics software system <a href="http://www.sagemath.org">SageMath</a>.
<p><a href="#top"><small>back to top</small></a>
<hr />
<!-- NEXT -->
<h2><code><a name="PG">Projective Geometry (Sabancı University)</code></h2>
<p>Graduate course</p>
<p>Contents: projective spaces over fields, collineations, correlations, perspectivities, elations, homologies, cross ratio, algebraic varieties, Klein correspondence, cubic surfaces, quadric, Hermitian varieties, polar spaces, incidence geometry, projective planes.
<p><a href="#top"><small>back to top</small></a>
<hr />
<h2><code><a name="LA">Linear Algebra (Sabancı University)</code></h2>
<p>Undergraduate course for engineering and computer science students</p>
<p>Contents: matrices, vector spaces, bases, linear independence, linear transformations, eigenvalues, eigenvectors, eigenspaces, diagonalisation, orthogonal matrices, orthonormal bases.
<p>This course aims to introduce basic concepts of linear algebra such as vector spaces, bases, linear transformations, eigenvalues and eigenspaces. The course gives students training to develop their mathematical skills, analytical and critical thinking abilities, their ability to apply these capabilities to practical problems, and to communicate their knowledge of these areas. Part of the theory is illustrated using the mathematics software system <a href="http://www.sagemath.org">SageMath</a>.
<p><a href="#top"><small>back to top</small></a>
<hr />
<h2><code><a name="Intro_Algebra">Introduction to Algebra (Sabancı University)</code></h2>
<p>Graduate course</p>
<p>Contents:
<p><a href="#top"><small>back to top</small></a>
<hr />
<h2><code><a name="FG">Finite Geometry (Sabancı University)</code></h2>
<p>Graduate course</p>
<p>Contents:
<p><a href="#top"><small>back to top</small></a>
<hr />
<h2><code><a name="FALG">Fondamenti di Algebra Lineare e Geometria (University of Padua) (in Italian)</code></h2>
<p>Undergraduate course for engineering students</p>
<p>Contents: complex numbers, polynomials, matrices, vector spaces, bases, linear independence, linear transformations, eigenvalues, eigenvectors, eigenspaces, diagonalisation, orthogonal matrices, orthonormal bases.
<p><a href="#top"><small>back to top</small></a>
<hr />
<h2><code><a name="LAFF">Linear Algebra and Finite Fields</a> (Ghent University)</code></h2>
<p>An elective course for the Master of Mathematical Informatics at Ghent University.</p>
<p>This course is a remediating course for the courses <i>Coding Theory and Cryptography</i> and <i>Computational Group Theory</i>. The aim is to:<br/>
1. make the students acquainted with the basic terminology of the theory of linear algebra, polynomials and equations over finite fields;<br/>
2. give the students a profound insight into the combinatorial aspects of projective and affine spaces over finite fields;<br/>
3. provide the students, by means of an assignment (project), with the necessary experience and skill to analyse problems characteristic for the area, and present the possible solutions in a well-founded and convincing way.
</p>
<p>Further details can be found on the electronic learning environment <a href="http://minerva.ugent.be">Minerva</a> (for which you need to login).</p>
<p><a href="#top"><small>back to top</small></a>
<hr />
<h2><code><a name="PG_Ghent">Projectieve Meetkunde</a> (Ghent University)</code></h2>
<p>General course for 2nd Bachelor in Mathematics at Ghent University.
<p>Position of the Course:
The students should apply their pre-knowledge on geometry from the courses Linear Algebra and Analytic Geometry to the study of projective spaces. The way of setting up the course, will stimulate the student to be skilled in handling more abstract mathematical reasoning.</p>
<p>Contents:
Starting with the general definition of a projective space over fields (including homogeneous coordinates, the theorems of Desargues and of Pappus, collineations, correlations and perspectivities) a detailed study is given of projective lines over fields (isomorphisms between projective lines, cross ratio, the theorem of the complete 4-gon, involutions). After that a general study of polarities will be treated, including classification and standard forms; emphasizing the polarities in projective spaces over the reals, the complex numbers and a finite field. In the special case of polarities in projective planes over finite fields, the study of conics and hermitian curves will be treated, as well as generalizations such as ovals, hyperovals and unitals. There is also a treatise on some polar spaces of low rank. The last chapter of the course contains an axiomatic study of projective and affine planes; the coordinatisation method; the relation with ternary rings; as well as some simple non-Desarguesian planes will be discussed.
<p><a href="#top"><small>back to top</small></a>
<hr />
<h2><code><a name="gentlessen">Finite semifields and related structures in finite geometry</a> (Ghent University)</code></h2>
These <a href="gentlessen.pdf">notes</a> consist of material for approximately four hours, depending on the level of the students.
<p>
The first part is aimed at understanding the connection between translation planes and spreads; the so-called Andr{\'e}-Bruck-Bose construction. Next we define the notion of semifields (first studied by L. E. Dickson (1906)) and use a vector space representation to obtain semifield spreads, after which we can apply the Andr{\'e}-Bruck-Bose construction to obtain semifield planes. We define the notion of isotopy and mention the important result proved by A. A. Albert
in 1960, which says that two semifields are isotopic if and only if the corresponding planes are isomorphic. All of this belongs to, let's say, the "classical" part of the notes. It gives us the necessary background to be able to deal with some more recent topics in Finite Geometry which have received quite a lot of attention during the last two decades.
We start this "modern" part with a particular configuration of subspaces which turns out to be equivalent to semifields, i.e., each such configuration can be used to construct a semifield and vice versa, each semifield gives rise to such a configuration. We conclude these notes with an example of a well-studied structure in finite geometry related to semifields, namely translation ovoids (sometimes called semifield ovoids) of the parabolic quadric in 4-dimensional
projective space. We show their correspondence with semifields by constructing them from a particular case of the configuration of subspaces mentioned above.
<p><a href="#top"><small>back to top</small></a>
<hr />
<h2><code><a name="CC">Coding Theory and Cryptography</a>
(University of Canterbury, Christchurch, New Zealand)</code></h2>
<p>
I gave part of this course during my visit to the
University of Canterbury as an Erskine Fellow. The other part was
given by Geertrui Van de Voorde. The course deals with the mathematical ideas underlying modern
cryptography, including algebra, number theory and probability
theory. The course covers basic coding theory, including Hamming codes, weight
enumerators, sphere-packing bound, Singleton bound, MacWilliams
identities, parity check matrix, dual codes, mutually orthogonal latin
squares, decoding algorithms. The second part is an
introduction to cryptography, starting with the discrete log problem.
<p><a href="#top"><small>back to top</small></a>
<hr />
<h2><code><a name="teaching_experience">Other teaching experience</a></code></h2>
<ul class="ul1">
<li style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times">2005-2006 (Technische Universiteit Eindhoven)</li>
<ul class="ul2">
<li style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times">Wiskunde 3 (exercise classes) </li>
</ul>
<li style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times">2000-2001 (Technische Universiteit Eindhoven)</li>
<ul class="ul2">
<li style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times">Algebra 2 (exercise classes)</li>
<li style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times">Linear Algebra and Linear Analysis 1 (exercise classes)</li>
</ul>
<li style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times">1999-2000 (Technische Universiteit Eindhoven)</li>
<ul class="ul2">
<li style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times">Algebra 2 (exercise classes)</li>
<li style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times">Linear Algebra and Linear Analysis 1 (exercise classes)</li>
</ul>
<li style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times">1998-1999 (Technische Universiteit Eindhoven)</li>
<ul class="ul2">
<li style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times">Algebra 2 (exercise classes)</li>
<li style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times">Graph Theory and Discrete Optimalisation (exercise classes)</li>
</ul>
<li style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times">1997-1998 (Technische Universiteit Eindhoven)</li>
<ul class="ul2">
<li style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times">Graph Theory and Discrete Optimalisation (exercise classes)</li>
</ul>
</ul>
<p><a href="#top"><small>back to top</small></a>
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